On the variation of the fractional mean curvature under the effect of C^{1, \alpha} perturbations

Previous Article
Short-time existence of the second order renormalization group flow in dimension three
DCDS Home
This Issue
Next Article
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains

December 2015, 35(12): 5769-5786. doi: 10.3934/dcds.2015.35.5769

On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

Matteo Cozzi ^1,

1.	Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Received March 2014 Published May 2015

Abstract
Related Papers

In this brief note we study how the fractional mean curvature of order $s \in (0, 1)$ varies with respect to $C^{1, \alpha}$ diffeomorphisms. We prove that, if $\alpha > s$, then the variation under a $C^{1, \alpha}$ diffeomorphism $\Psi$ of the $s$-mean curvature of a set $E$ is controlled by the $C^{0, \alpha}$ norm of the Jacobian of $\Psi$. When $\alpha = 1$ we discuss the stability of these estimates as $s \rightarrow 1^-$ and comment on the consistency of our result with the classical framework.

Keywords: $C^{1, Fractional mean curvature, \alpha}$ diffeomorphisms, fractional Laplacian, quantitative Implicit Function Theorem., non-local equations, $s$-perimeter.

Mathematics Subject Classification: Primary: 35R11, 28A75, 49Q05; Secondary: 26B1.

Citation: Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769

References:

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4.
[2]	N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815. doi: 10.1080/01630563.2014.901837.
[3]	G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238. doi: 10.1016/j.jmaa.2015.04.002.
[4]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.
[5]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284. doi: 10.1017/S0956792598003453.
[6]	B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639.
[7]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.
[9]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[11]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[12]	L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.
[13]	L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6.
[14]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871. doi: 10.1016/j.aim.2013.08.007.
[15]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[16]	S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790. doi: 10.3934/dcds.2013.33.2777.
[17]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015). doi: 10.1515/crelle-2015-0006.
[18]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.
[19]	M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.
[20]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.
[21]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.
[22]	G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.
[23]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500. doi: 10.1016/j.anihpc.2012.01.006.
[24]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39. doi: 10.1007/s00526-012-0539-7.
[25]	E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23. doi: 10.1007/s00032-013-0199-x.

show all references

References:

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4.
[2]	N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815. doi: 10.1080/01630563.2014.901837.
[3]	G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238. doi: 10.1016/j.jmaa.2015.04.002.
[4]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.
[5]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284. doi: 10.1017/S0956792598003453.
[6]	B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639.
[7]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.
[9]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[11]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274.
[12]	L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.
[13]	L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240. doi: 10.1007/s00526-010-0359-6.
[14]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871. doi: 10.1016/j.aim.2013.08.007.
[15]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[16]	S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790. doi: 10.3934/dcds.2013.33.2777.
[17]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015). doi: 10.1515/crelle-2015-0006.
[18]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.
[19]	M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.
[20]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.
[21]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.
[22]	G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718. doi: 10.1007/s10231-011-0243-9.
[23]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500. doi: 10.1016/j.anihpc.2012.01.006.
[24]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39. doi: 10.1007/s00526-012-0539-7.
[25]	E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23. doi: 10.1007/s00032-013-0199-x.

[1]	Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
[2]	Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022021
[3]	Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155
[4]	Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[5]	Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3387-3399. doi: 10.3934/dcdss.2021017
[6]	Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014
[7]	Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
[8]	Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236
[9]	Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[10]	Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
[11]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[12]	Bassam Fayad, Zhiyuan Zhang. An effective version of Katok's horseshoe theorem for conservative C² surface diffeomorphisms. Journal of Modern Dynamics, 2017, 11: 425-445. doi: 10.3934/jmd.2017017
[13]	Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139
[14]	Tianling Gao, Qiang Liu, Zhiguang Zhang. Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021254
[15]	Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
[16]	Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319
[17]	Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154
[18]	A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35
[19]	Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2515-2535. doi: 10.3934/dcdsb.2021146
[20]	Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Matteo Cozzi
on Google Scholar
Matteo Cozzi

[Back to Top]